Problem: Simplify the following expression: $k = \dfrac{-63q^2 - 27q}{27q}$ You can assume $q \neq 0$.
Solution: Find the greatest common factor of the numerator and denominator. The numerator can be factored: $-63q^2 - 27q = - (3\cdot3\cdot7 \cdot q \cdot q) - (3\cdot3\cdot3 \cdot q)$ The denominator can be factored: $27q = (3\cdot3\cdot3 \cdot q)$ The greatest common factor of all the terms is $9q$ Factoring out $9q$ gives us: $k = \dfrac{(9q)(-7q - 3)}{(9q)(3)}$ Dividing both the numerator and denominator by $9q$ gives: $k = \dfrac{-7q - 3}{3}$